Complete linear Weingarten surfaces of Bryant type. A Plateau problem at infinity.

Complete linear Weingarten surfaces of Bryant type. A Plateau problem at infinity. José Antonio Gálvez Antonio Martínez Francisco Milán Abstract In this paper we study a large class of Weingarten surfaces which includes the constant mean curvature one surfaces and flat surfaces in the hyperbolic 3-space. We show that these surfaces can be parametrized by holomorphic data like minimal surfaces in the Euclidean 3-space and we use it to study their completeness. We also establish some existence and uniqueness theorems by studing the Plateau problem at infinity: when is a given curve on the ideal boundary the asymptotic boundary of a complete surface in our family? and, how many embedded solutions are there? Key words and phrases: Hyperbolic 3-space, Weingarten surfaces, Plateau problem, Weierstrass data. 1 Introduction. During the last two decades there has been a renewed interest in the geometry of surfaces in hyperbolic manifolds. Basic problems such as existence, symmetry, behaviour at infinity, global structure and classification of complete surfaces of constant curvature in the hyperbolic 3-space H 3 have called the attention of several authors see for example, [5], [11], [12], [18]. Concerning constant mean curvature one surfaces in H 3, also called Bryant surfaces, it was crucial the paper of R. Bryant [2], who showed how to parametrize these surfaces by meromorphic data and began the study of their geometry. Since Bryant s work many properties and examples have been discovered in [3], [15], [22], [25], [26] among others. By considering the complex structure determined by the second fundamental form, the authors proved in [6] that a holomorphic resolution like the Bryant representation also holds for Research partially supported by MCYT-FEDER Grant No. BFM2001-3318 and Junta de Andalucía CEC: FQM0203. 2000 Mathematics Subject Classification. Primary: 53C42. Secondary: 53A35 1

flat surfaces in H 3 and used it to the study of some examples and the behaviour at infinity of complete ends. The most important tool in these representations is that both kinds of surfaces have a conformal hyperbolic Gauss map with respect to a natural complex structure determined by either the first, I, or the second, II, fundamental form. In this paper we extend the two above mentioned Weierstrass type representations through the investigation of Weingarten surfaces. We shall study surfaces in H 3 whose mean curvature H and Gauss curvature K I satisfy a linear relation of the form 2aH 1 + bk I = 0, for some a, b R, of elliptic type, i. e. a + b 0. We call them linear Weingarten surfaces of Bryant type, in short, BLW-surfaces. These surfaces are a particular case of the special surfaces studied by Sa Earp and Toubiana in [21] see also [17], [19] and [20]. A BLW-surface Σ may be considered lying on a straight half-line λ ab in the H, K I -plane, starting at the point 1, 0: λ ab = {1 tb, 2ta : t 0}, see Figure 1 and Remark 3. The geometry of Σ is rather different depending on whether Σ is in one of the following regions, R 1 = {H, K I : 1 H, 0 K I }, R 2 = {H, K I : 0 K I, 2H 1 K I }, R 3 = {H, K I : 2H 1 K I 0}, R 4 = {H, K I : 2H 1 K I, 0 K I }. If Σ is in λ ab, we shall see that σ = ai + bii is a Riemannian metric which induces a natural complex structure on Σ such that its hyperbolic Gauss map is a conformal map. We shall prove that the complex representations of Bryant [2] and the authors [6] extend to BLW-surfaces see Theorem 2. The previous complex resolution is applied to the study of completeness. Thus, we shall start by proving Theorem 3 that a complete BLW-surface in R 1 R 2 is a totally umbilical round sphere, a horosphere or a hyperbolic cylinder. The regions R 3 and R 4 are much richer in global examples. Every complete BLW-surface Σ in λ ab R 4, with meromorphic data h, ω has an associated complete constant mean curvature one surface, Σ 0, with geometry as Σ and with meromorphic data a/a + b h, ω. Concerning R 3, we shall see that complete BLW-surfaces in λ ab R 3, are conformally equivalent to the unit disk, D, and we shall prove Theorem 4 that they are, up to conformal transformations of D, in correspondence with the set of meromorphic maps G : D C { } with bounded Schwarzian derivative. In that way the hyperbolic Gauss map and the standard Euclidean Gauss map of complete BLW-surfaces in the half space model of H 3 lying on R 3 give geometric meaning to classical families of complex functions which have been studied in connection with the Schwarzian derivative see, for example, [9], [13], [14], [16]. We also establish some existence and uniqueness theorems for complete BLW-surfaces in R 3. Our main interest shall be the Plateau problem at infinity: when is a given curve on the ideal boundary S 2 of H 3 the asymptotic boundary of a complete BLW-surface? 2

If λ ab R 3, we shall prove Theorem 5 that a convex Jordan curve Γ S 2 is the asymptotic boundary of a unique embedded complete BLW-surface in λ ab with an inner hyperbolic normal. Moreover if 1/2 < a/a + b < 0 there are exactly two embedded complete BLW-surfaces on λ ab with asymptotic boundary Γ. Finally, we shall deal with the general case and shall prove Theorem 6 that if 1/2 < a/a + b < 0, a Jordan curve Γ on S 2 is the asymptotic boundary of at least two complete BLW-surfaces on λ ab. If 1/4 < a/a + b < 0, then there only exist two embedded complete BLW-surfaces on λ ab with asymptotic boundary Γ. 2 Conformal representation. In the Lorentz-Minkowski model, L 4, the hyperbolic 3-space is described by one connected component of a two-sheeted hyperboloid. More precisely, { } H 3 = x 0, x 1, x 2, x 3 R 4 : x 2 0 + x 2 1 + x 2 2 + x 2 3 = 1, x 0 > 0 endowed with the induced metric of L 4 given by the quadratic form x 2 0 + x2 1 + x2 2 + x2 3. Associated with that metric we shall denote by the usual exterior product in H 3, that is, 1 u v = p u v, for any tangent vectors u, v at a point p H 3, where p u v is the unique vector in L 4 such that p u v, w = detp, u, v, w, for any w L 4. Here.,. denotes the inner product in L 4 and det the usual determinant. If we consider { } N 3 + = x 0, x 1, x 2, x 3 R 4 : x 2 0 + x 2 1 + x 2 2 + x 2 3 = 0, x 0 > 0 the positive null cone in L 4, then N 3 +/R + inherits a natural conformal structure and it can be regarded as the ideal boundary S 2 of H3. We shall regard L 4 as the space of 2 2 Hermitian matrices, Herm2, in the standard way see [2], [6], by identifying x 0, x 1, x 2, x 3 L 4 with the matrix x0 x 3 x 1 + ix 2. x 1 ix 2 x 0 + x 3 Under this identification one has m, m = detm for all m Herm2. Then H 3 is the set of m Herm2 with detm = 1. The action of SL2, C on these Hermitian matrices defined by g m = gmg, where g SL2,C, g = t g, preserves the inner product and leaves H 3 invariant. 3

The space N 3 + is seen as the space of positive semi-definite 2 2 Hermitian matrices of determinant 0 and its elements can be written as w t w, where t w = w 1, w 2 is a non zero vector in C 2 uniquely defined up to multiplication by a unimodular complex number. The map w t w [w 1, w 2 ] CP 1 becomes the quotient map of N 3 + on S2 and identifies S2 to CP 1. Thus, the natural action of SL2, C on S 2 is the action of SL2, C on CP1 by Möbius transformations. Now, let S be a surface and ψ : S H 3 an immersion with Gauss map η. Then, ψ is called a linear Weingarten immersion of Bryant type, in short, BLW-surface if the mean curvature H and the Gauss curvature K I satisfy a linear relation of the form, 2 2 a H 1 + b K I = 0, for some a, b R, a + b 0. Remark 1 This particular kind of linear Weingarten immersions includes the Bryant surfaces and the flat surfaces. The immersions satisfying 2 with a + b = 0, that is, the non elliptic case, are the ones with a constant principal curvature equal to 1. They are studied in [1]. Lemma 1 Let ψ : S H 3 be a BLW-surface. Then, we can consider that a + b = 1, 3 2 a H 1 + b K 1 = 0 and σ = a I + b II is a positive definite metric, where K is the Gauss-Kronecker curvature, I = dψ, dψ and II = dψ, dη, the first and second fundamental form of the immersion, respectively. Proof : Since K I = K 1, by scaling a and b in 2, we can assume that 3 is satisfied, with a + b = 1. Let {e 1, e 2 } be an orthonormal basis at a point p which diagonalizes dη, that is, dηe i = k i e i, i = 1, 2. Then, 4 σe 1, e 1 σe 2, e 2 σe 1, e 2 2 = a + bk 1 a + bk 2 = = a 2 + b 2aH + bk = a + b 2 = 1 and so σ is definite. Now, by changing the sign of a and b if necessary, we obtain that σ is positive definite and a + b = 1, which concludes the proof. From now on, we will suppose that every BLW-surface satisfies the above lemma and we shall regard S as a Riemann surface with the conformal structure induced by σ = a I + b II. 4

Lemma 2 Let ψ : S H 3 be a BLW-surface and z be a conformal parameter for σ. Then 5 aψ z bη z = iη ψ z, where, for instance, ψ z denotes the derivative of ψ with respect to z. Proof : Let z be a conformal parameter such that σ = 2ρ dz 2. From 4, 6 ψ z ψ z, η = iρ. Since aψ z bη z is a tangent vector field, it can be written as for some local complex functions α and β. Now, using that z is conformal and 6, aψ z bη z = αη ψ z + βη ψ z 0 = σψ z, ψ z = aψ z bη z, ψ z = β η ψ z, ψ z = iβρ, ρ = σψ z, ψ z = aψ z bη z, ψ z = α η ψ z, ψ z = iαρ and we are done. As in Bryant s paper [2], we consider for any immersion ψ : S H 3 with associated Gauss map η its hyperbolic Gauss map as the map given by G := [ψ + η]. That is, for any p S, the oriented geodesic emanating from p meets the ideal boundary S 2 of H 3 at Gp. Theorem 1 Let ψ : S H 3 be a BLW-surface. Then ψ + η is a conformal map with respect to the metric σ = ai + bii and 7 σ ψ + η = 2 {H 1ψ + K Hη}, a + b where σ denotes the Laplacian of σ. In particular, its hyperbolic Gauss map G is conformal. Moreover, the immersion lies in a horosphere or dψ + η, dψ + η is a pseudometric of constant curvature a a+b. Proof : Let z be a conformal parameter such that σ = 2ρ dz 2. Then we can write η z = Λψ z + Υψ z, η z = Υψ z Λψ z for a local real function Λ and a complex function Υ. 5

Thus, H = Λ and K = Λ 2 Υ 2 = H 2 Υ 2, that is, 8 η z = Hψ z + Υψ z. Therefore, using 3 a + bh 2 b 2 Υ 2 = a 2 + 2abH + b 2 K = a + b 2 = 1. The cross product of 5 and 8 with ψ z and ψ z gives { bηz ψ z = iη ψ z ψ z = i ψ z, ψ z η, bη z ψ z = bυψ z ψ z, { aψz bη z ψ z = iη ψ z ψ z = i ψ z, ψ z η, aψ z bη z ψ z = aψ z ψ z + bhψ z Υψ z ψ z = a + bhψ z ψ z, and by taking the inner product with η one has from 6 9 ψ z, ψ z = bυρ, ψ z, ψ z = a + bhρ. In that way, from 5, 10 bψ + η z = a + bψ z + iη ψ z and, using 9, one obtains ψ + η z, ψ + η z = 0 when b 0. Moreover, if b = 0 then, from 3 and 8, we have that H = 1, ψ + η z = Υψ z and, using 9, ψ + η z, ψ + η z = 0. Therefore, ψ + η is a conformal map in any case. Now, bearing in mind 1 and 6, if we take the derivative of 5 with respect to z, then aψ z z bη z z = ρψ ihψ z ψ z iψ η ψ z z. As the above equality has imaginary part 0 = ψ η ψ z z, we have 11 aψ z z bη z z = ρψ + Hη. On the other hand, 5 also gives aψ z bη z η = iψ z and by deriving with respect to z we obtain 12 ψ z z = ρ{a + bhψ + ah + bkη}. If b 0, then 12 minus 11 respectively, 12 plus 11 and 6 give the expression 7 when a + b = 1 respectively, when a + b = 1. If b = 0, then H = 1, σ = I, ψ z, ψ z = 0 and ψ z, ψ z = ρ, hence ψ zz = ρ z ρ ψ z Υρη, ψ zz = ρη, η z = ψ z + Υψ z. 6

Since ψ zz z = ψ zz z, we have ρ Υ z = 0 and, consequently, ψ + η z z = Υψ z z = Υ 2 ρη = ρk 1η. In order to determine the singular points of ψ + η, we have from 6, 9 and 10 that 13 b 2 ψ + η z, ψ + η z = a + bψ z + iη ψ z, a + bψ z iη ψ z = = 2 ψ z, ψ z a + bρ = 2bH 1ρ. And we conclude that if b 0, ψ + η has a singular point at z 0 if and only if Hz 0 = 1. Moreover, from 3 and 9, if Hz 0 = 1 and b 0 then Kz 0 = 1 and a + b = 1. When b = 0 the same conclusion is clear from 8, with a = 1 because σ is positive definite. Now, from 2 and 7, ψ can be written as 14 ψ = a + b ρ2h K 1 ψ + η z z + 2a + b ψ + η, 2a + b at the regular points of ψ + η. Using that ψ +η is conformal, we have that there exist holomorphic functions, A, B, such that [ψ + η] is represented as [A, B] CP 1 S 2 and 15 ψ + η = λ A B AA AB A, B = λ AB BB for some positive regular function λ. Thus, 14 gives ψ = gωg and η = g Ωg,, with A Az δλz z βλ δλ z 1 + βλ δλz z δλ z g = B B z, Ω = δλ z δλ, Ω = δλ z δλ, β = 2a+b 2a+b and δ = and, consequently, a+b ρ2h K 1. From these expressions one gets 2 = η, η ψ, ψ = detη + detψ = δλ 2 detg 2, 1 = ψ, ψ = detψ = 2β + logλ 2 detg 2 z z = 1 + 2β λ 2 detg 2 = 2 7 4 λ 4 detg 2 λλ z z λ z λ z a 2a + b λ2 detg 2.

a a+b at That is, dψ + η, dψ + η = λ 2 detg 2 dz 2 has constant Gauss curvature ε = the regular points of ψ + η. From 7 and 12, the two 2-forms Q I = ψ z, ψ z dz 2 and Q II = ψ z, η z dz 2 are holomorphic on S. And, since ψ has an umbilical point at z 0 if and only if Q I z 0 = 0 = Q II z 0, then ψ is totally umbilical or has isolated umbilical points. Hence, either ψ is totally umbilical with H = 1 = K, that is, ψs lies on a horosphere or dψ + η, dψ + η is a pseudometric of constant curvature ε. Remark 2 If ψ : S H 3 is a BLW-surface and S is a topological sphere then, using that Q I and Q II are holomorphic, they vanish identically and we obtain that ψs is a totally umbilical round sphere. Remark 3 The regions R 1, R 2, R 3 and R 4 given in the introduction separate into four parts the exterior of the domain H 2 K I + 1 < 0 in the H, K I -plane according to the sign of a + b and ε = a/a + b, see Figure 1. In fact, if b 0, from 7 and 13 we obtain 2 b H 1 = 1 ρ ψ + η z, ψ + η z = 1 ρ ψ + η zz, ψ + η = 2H 1 K I. a + b When a + b = 1, from 3, that is, 2H 1 K I = a + bk I = ak I 2aH 1 = ak I 2H 1, 1. if a 0 then 2H 1 K I, K I 0 and we are in the region R 4, 2. if a 0 then 2H 1 K I 0 and we are in the region R 1. In a similar way, when a + b = 1 we are either in R 2 or in R 3. Moreover, from 13, if b 0 it is satisfied that H 1 and if b 0 then H 1. Consequently, from 3, if a 0 one gets K I 0 and if a 0 then K I 0. Theorem 2 Conformal representation i Let S be a non compact, simply connected surface and ψ : S H 3 a BLW-surface. Then, there exist a meromorphic curve g : S SL2, C and a pair h, ω consisting of a meromorphic function h and a holomorphic 1-form ω on S, such that the immersion and its Gauss map can be recovered as 16 ψ = gωg and η = g Ωg, where 17 1+ε 2 h 2 Ω = 1+ε h ε h 2 εh 1 + ε h 2 1 ε 2 and Ω h 2 = 1+ε h ε h 2 εh 1 ε h 2, 8

a with ε = a+b and 1 + ε h 2 > 0. Moreover, the curve g satisfies g 1 0 ω 18 dg =. dh 0 The induced metric and σ = a I + b II are given, respectively, by 1 ε 2 dh 2 19 I = 1 εωdh + 1 + ε h 2 2 + 1 + ε h 2 2 ω 2 + 1 ε ωd h and 20 σ = a + b 1 + ε h 2 2 ω 2 1 ε2 dh 2 1 + ε h 2 2 ii Conversely, let S be a Riemann surface, g : S SL2, C a meromorphic curve and h, ω a pair as above satisfying 18 and such that 20 is a positive definite metric. Then ψ = gωg : S H 3, Ω as in 17, is a BLW-surface satisfying 3 with induced metric and σ given by 19 and 20. Proof : Let us assume that ψs is not a piece of a horosphere, otherwise the result is easily obtained with h constant. Since S is non compact and simply connected there exist a global conformal parameter z and as dψ + η, dψ + η is a pseudometric on S of constant Gauss curvature ε then, from the Frobenius theorem, there exists a meromorphic function h on S holomorphic if ε 0 such that 1 + ε h 2 > 0 and dψ + η, dψ + η = λ 2 detg 2 dz 2 =. 4 dh 2 1 + ε h 2 2, where we follow the same notation as above. From this expression, if h has a pole of order m, then dh has a pole of order m + 1 and detg has a zero of order m 1. Hence, there exists a meromorphic function R on S holomorphic if ε 0 with 21 R 2 = dh detgdz 0 and ARBR z BRAR z = h z. Now, if we replace A and B by AR and BR, respectively, in 15 and choose the matrix A 1 g = h z A z 22 B 1 h z B z with meromorphic coefficients and detg=1 on S, then the new λ 2 is 4/1+ε h 2 2. Moreover, 0 g 1 1 dg = h A 2 zz B z A z B zz z dz h z 0 9

and we obtain 18 for ω = AzzBz AzBzz h dz. 2 z From 14 one has that 16 is satisfied for Ω = δλz z βλ δλ z h z δλ z h z δλ h z 2 and Ω = 1 + βλ δλz z δλ z h z δλ z h z δλ h z 2 where 2 = η, η ψ, ψ = δλ 2 h z 2. Therefore, 16 and 17 are satisfied. Besides, 0 ω 0 dh dψ = dgωg = g Ω + dω + Ω g dh 0 ω 0 = and = g dη = dg Ωg = g = g 1 + ε h 2 ω + 1 ε 1+ε h dh 2 1 + ε h 2 ω + 1 ε 1+ε h dh 2 0 0 ω dh 0 Ω + d Ω + Ω 0 dh ω 0 g = 1 + ε h 2 ω + 1+ε 1+ε h dh 2 1 + ε h 2 ω + 1+ε 1+ε h dh 2 0 dψ + η = g 2dh 1+ε h 2 2dh 1+ε h 0 2 g. Hence, one calculates the induced metric I, 19, and the metric σ = ai + bii, 20, from I = dψ, dψ = detdψ g, g and II = dψ, dη = 1 detdψ + η detdψ detdη. 2 In particular, 1 + ε h 2 2 ω 2 is a real expression on S and consequently ω must be holomorphic. Following the same notation as in [2] and [6] the pair h, ω given by the above theorem will be called the Weierstrass data associated with the conformal representation of the BLW-surface ψ. However, from 18 it is clear that the pair h, g could be also called the Weierstrass data of the immersion. 10

Remark 4 If S is a non compact, simply connected surface and ψ 1, ψ 2 : S H 3 are two BLW-surfaces satisfying 3 with the same Weierstrass data h, ω, then ψ 1 and ψ 2 are the same immersion, up to a rigid motion in H 3. Indeed, ψ 1 = g 1 Ωg 1 and ψ 2 = g 2 Ωg 2 where g 1 1 dg 1 = 0 ω dh 0 = g 1 2 dg 2 for some meromorphic curves g 1, g 2 : S SL2, C. Thus, one has dg 1 g2 1 = dg 1 g2 1 + g 1 dg2 1 0 ω = g 1 dh 0 g 1 2 + g 1 0 ω dh 0 g 1 2 = 0 and there exists a constant matrix g 0 SL2, C such that g 1 = g 0 g 2. Consequently, ψ 1 = g 0 ψ 2 g0 as we wanted to prove. It is interesting to observe that if ψ : S H 3 is an immersion with H = 1 then ε = 1, 0 i 0 i Ω = i ih i i h and ψ = gωg = F F, where 0 i F = g i ih SL2, C, that is, the conformal representation becomes the Bryant s representation see [2], [25]. Moreover, if the immersion does not lie in a horosphere and we denote by G its hyperbolic Gauss map then taking A = G and B = 1 in 15 one gets from 21 and 22 g = ig dh dg i dh dg i dh d G i dh d dh dg dh dg Thus, by considering the new parameter ζ = h where that is possible, one has F = 1 2G 2 ζ G G ζζ 2G ζ G ζg ζ + ζ G G ζζ 2G 3/2, G ζζ 2G ζ + ζ G ζζ ζ that is, we recover the Small s formula for surfaces with constant mean curvature one see [10], [23].. 11

3 Completeness of the immersions. From Remark 3 it is clear that the Gauss curvature K I of a BLW-surface ψ : S H 3 is non negative on S or non positive everywhere. Moreover, if S is non compact then, from 16, 17, 19 and 20, its Gaussian curvature K I = K 1 can be calculated as 23 K I = 4 ε dh 2 1 + ε h 2 4 ω 2 1 ε 2 dh 2 and its mean curvature is given by H = 1 + 2 1 ε dh 2 1 + ε h 2 4 ω 2 1 ε 2 dh 2. In particular, a point p S is umbilical if and only if dhp = 0 or ωp = 0. We shall study separately the cases of complete immersions with non positive and non negative Gauss curvature. 3.1 Completeness with non negative Gauss curvature. First, we give some examples of complete BLW-surfaces and then we classify them as the only non totally umbilical complete BLW-surfaces with non negative Gauss curvature. Let us consider S = C, hz = z + c 0 and ωz = c 1 dz for some complex constants c 0 and c 1 : 1. If we take a = 0 and b = 1 then, from Theorem 2, one obtains the flat immersion 24 ψu, v = 1+r 2 1 r2 2r e u 2r e iv 1 r 2 1+r2 2r eiv 2r eu with r = c 1, u + iv = 2 c 1 z and σ = c 1 2 1 dz 2 positive definite when c 1 > 1. Moreover, the induced metric I = c 1 dz 2 + 1 + c 1 2 dz 2 + c 1 d z 2 c 1 1 2 dz 2 is complete. 2. If we put a = 0 and b = 1 then 24 is a complete flat immersion with 0 < c 1 < 1. In any case, the immersion given by 24 may be regarded as the set of points at a fixed distance from the geodesic e t 0 γt = 0 e t, namely, a hyperbolic cylinder., 12

Theorem 3 Let ψ : S H 3 be a complete BLW-surface with non negative Gauss curvature K I. Then ψs is a totally umbilical round sphere, a horosphere or a hyperbolic cylinder. Proof : We can assume that S is simply connected, otherwise we would pass to the universal covering surface of S. Moreover, if S is compact then, from Remark 2, ψs is a totally umbilical round sphere. First, let us suppose that ψ is a non flat immersion, that is, there exists a point on S such that K I > 0 in particular, from Remark 3, a < 0. If a+b = 1 then from 3 and since H 2 K, we have that K I 4a/b 2 > 0. Therefore, from Bonnet s theorem, S must be compact. If a + b = 1, ε < 0 and from 19 1 2 I τ = 1 ε2 dh 2 1 + ε h 2 2 + 1 + ε h 2 2 ω 2 and τ must be a complete metric on S. Besides, using that σ is a positive definite metric and that the bounded holomorphic function h satisfies that 1 + ε h 2 < 1 we obtain τ 21 + ε h 2 2 ω 2 2 ω 2. Therefore, 2 ω 2 is a flat complete metric conformal to σ. Thus, S is conformally equivalent to the complex plane C and h must be constant, that is, ψs is a horosphere. On the other hand, if ψ is a flat immersion then a = 0 = ε and from 19 τ = dh 2 + ω 2 is a complete metric. Since σ is a positive definite metric one has that if b = 1 respectively, b = 1 then τ 2 ω 2 resp. τ 2 dh 2 and, hence, there exists a complete conformal flat metric on S. Thus, S is conformally equivalent to C and the function dh/ω or ω/dh is constant with modulus less than one when b = 1 or b = 1. Therefore, ψs is a horosphere or ω = c 1 dh for a complex constant c 1 0 with c 1 1. Thus, in the second case dh 0 everywhere because σ is non degenerate and we can take the new conformal parameter ζ such that dζ = dh. Then one obtains hζ = ζ + c 0 and ωζ = c 1 dζ for a constant c 0 C and ψ lies on a hyperbolic cylinder. 3.2 Completeness with non positive Gauss curvature. A Plateau problem at infinity. Now, we focus our attention on the study of the completeness of the immersions with non positive Gauss curvature. We remark that given a simply connected surface S and a non flat BLW-surface ψ : S H 3 with non positive Gauss curvature then S cannot be compact and a > 0. Therefore, the immersion lies in the set R 3 or R 4. In the last case, the geometry of the surface is similar to the one of a surface with constant mean curvature H = 1. In fact, if h, ω are the Weierstrass data for a BLW-surface ψ 0 in 13

R 4 then ε > 0 and 1 + ε h 2 ω 2 is a positive definite metric because so is σ. Moreover, we obtain a new associated immersion ψ 1 with constant mean curvature one and Weierstrass data εh, ω. If ψ 0 is a complete immersion then, from 19 and 20, 1 + ε h 2 ω 2 is also a complete metric, that is, ψ 1 is a complete immersion. Thus, the study of complete BLW-surfaces in R 4 can be reduced to the study of complete immersions with constant mean curvature one. Many things are known in this case and some very interesting results were proved in [2], [3] and [25]. However, the geometry of the surface is very different when the immersion lies in R 3. For instance, Lemma 3 Let ψ : S H 3 be a complete BLW-surface in R 3 with Weierstrass data h, ω. Then, S is conformally equivalent to the unit disk D and h is a global diffeomorphism onto D ε = {z C : z 2 < 1/ε}. Proof : Since the induced metric is complete and σ is positive definite, from 19 and 20 1 2 I 1 ε2 dh 2 1 + ε h 2 2 + 1 + ε h 2 2 ω 2 2 1 ε2 dh 2 1 + ε h 2 2 and the metric 4 dh 2 /1 + ε h 2 2 must be complete. Therefore, h : S D ε is bijective and S is conformally equivalent to the unit disk. From the above lemma, given a complete BLW-surface ψ : S H 3 with Weierstrass data h, ω we can consider, up to a change of parameter, S = D and hz = z/ ε. Moreover, from 20, σ is positive definite if and only if 25 ω < 1 ε ε dz 1 z 2 2, z D. On the other hand, if we consider the hyperbolic Gauss map of the immersion G and take A = G, B = 1 in 15 one gets from 21 and 22 26 Thus, g = ig dh dg i dh dg g 1 dg = i dh d G i dh d dh dg dh dg 0 = ig 4 ε Gz i 4 ε Gz ε 3G 2 zz 2G zg zzz 4G 2 z 1 ε 0 i 4 G ε Gz i 4 1 ε Gz dz, z z. 14

and from 18 and 25, the Schwarzian derivative of G satisfies {G, z} := d Gzz 1 2 Gzz = 2G zg zzz 3G 2 zz dz 2 and G z {G, z} < G z 21 ε ε 2G 2 z 1 1 z 2 2 z D. = 2 ε ω dz Consequently, G is a local diffeomorphism with bounded Schwarzian derivative. Otherwise, there would exist a point z 0 such that G is not locally one to one and we could write G z in a neighbourhood of z 0 as G z = z z 0 k with c 0 0 and an integer k 0. Hence kk + 2 {G, z} = 2 n=0 c n z z 0 n 1 z z 0 2 k c 1 2 c 0 1 z z 0 + hz where h is a holomorphic function in a neighbourhood of z 0. But, either {G, z} = at z 0 which contradicts the above inequality or k = 2 and G has a pole of order one at z 0, that is, G is locally one to one. The above considerations are summarized in the following result: Theorem 4 Let S be a simply connected surface and ψ : S H 3 a complete BLW-surface in R 3. Then one has i S can be identified with D and its Weierstrass data h can be taken as hz = z/ ε, ii the hyperbolic Gauss map G : D C { } is a local diffeomorphism satisfying 27 {G, z} < 21 ε ε 1 1 z 2 2, z D, iii the immersion can be recover as 1 G 2 ε GA + Gz 1 z 2 2 1 ε A 2 G + ε G z 1 z 2 A 28 ε Gz 1 z 2 1 ε A 2 G + ε G z 1 z 2 A 1 ε A 2 and its Gauss map η as 1 G 2 + ε GA + Gz 1 z 2 2 1 + ε A 2 G ε G z 1 z 2 A 29 ε Gz 1 z 2 1 + ε A 2 G ε G z 1 z 2 A 1 + ε A 2 with A = G zz 2G z 1 z 2 z. 15

Conversely, let G : D C { } be a meromorphic map. Then, if G satisfies 27 one has that 28 is a BLW-surface in R 3 with hyperbolic Gauss map G and Weierstrass data z/ ε, 1 2 ε {G, z} dz. Moreover, if 30 {G, z} with b 0 < 21 ε ε, then the immersion is complete. b 0 1 z 2 2 z D, Proof : Let ψ : S H 3 be a complete BLW-surface in R 3 and S simply connected. Then, we can assume that i and ii are satisfied and, following Theorem 2, the immersion and its Gauss map can be recover as in 16 and 17 where g is given as in 22 for some meromorphic functions A and B. Thus, 1 A ψ = 2 εa 1 A 1 AB εa 1 A 2 1 z 2 AB εa 1 A 2 B 2, εa 2 A 2 where 31 A 1 = A z + A z 1 z 2, A 2 = B z + B z 1 z 2. Now, using that G = A/B, g SL2, C, that is, εab z BA z = 1 and one has G z = 1 ε B 2, G zz G z = 2 B z B, A 2 = B z + B z B 1 z 2 = BA, 1 A 1 = GA 2 1 z 2 = B GA + G z 1 z 2, ε B and bearing in mind that ε B 2 G z = 1 from 31, one obtains 28. Analogously, one can compute 29. From Theorem 2, the converse part is a straightforward computation taking g as in 26. Moreover, if G satisfies 30, the induced metric I can be estimated as follows I = 1 ε dh 2 1 ε dh 2 = 1 + ε h 2 + 1 + ε h 2 ω 1 + ε h 2 1 + ε h 2 ω 2 ε 1 ε ε {G, z} dz 2 2 ε 2 b 0 1 ε ε 1 z 2 1 z 2 Since b 0 < 21 ε ε 2 dz 2 1 z 2 2., then 1 ε ε ε 2 b 0 is a positive number and the induced metric is, up to a constant, greater than or equal to the Poincaré metric on the unit disk. Hence, the immersion is complete. 16

Remark 5 Let ψ 0 : D H 3 be a BLW-surface in R 3 with Weierstrass data h 0 z = z/ ε and hyperbolic Gauss map G 0. If we consider the new immersion ψ 1 : D H 3 with h 1 ζ = ζ/ ε and G 1 ζ = G 0 ϕζ where ϕ : D D is a conformal equivalence, then ψ 0 ϕζ = ψ 1 ζ. Indeed, since ϕ is a biholomorphic map onto the unit disk for some ζ 0 D, θ R and Thus, one has ϕζ = e iθ ζ + ζ 0 ζ 0 ζ + 1, ζ D, ϕ ζ ζ 1 ϕζ 2 = e iθ 1 + ζ 0ζ 1 + ζ 0 ζ 1 1 ζ 2. G 1 ζ ζ 1 ζ 2 = e iθ 1 + ζ 0ζ 1 + ζ 0 ζ G 0 zϕζ 1 ϕζ 2 A ψ1 ζ = e iθ 1 + ζ 0ζ 1 + ζ 0 ζ A ψ 0 ϕζ and the above equality is clear from 28. Now, for the representation of some surfaces and the study of its geometric behaviour, we will identify the hyperbolic space H 3 L 4 with the upper half space of R 3 given by {x 1, x 2, x 3 R 3 : x 3 > 0} by considering the map 32 x 0, x 1, x 2, x 3 1 x 0 + x 3 x 1, x 2, 1. Under this identification the hyperbolic metric is given by 33 ds 2 = 1 x 2 3 dx 2 1 + dx2 2 + dx2 3 and its ideal boundary S 2 is identified with the one point compactification of the plane Π {x 3 = 0}, that is, S 2 = Π { }. Thus, the asymptotic boundary of a set Σ H 3 is Σ = clσ S 2 where clσ is the closure of Σ in {x 3 0} { }, that is, in the one point compactification of H 3 Π. 17

Example 1 Let ψ : D H 3 be a complete totally umbilical BLW-surface with ε < 0. From the comments at the beginning of Section 3 and from Lemma 3, we have that the Weierstrass data of the immersion are z/ ε, 0, z D. In that case Gz = rz, r > 0 is a holomorphic solution of {G, z} = 0. Consequently, from Theorem 4 and 32 the immersion is isometric to the following one: ψ 1 u, v = r 1 εu 2 + v 2 {1 εu, 1 εv, ε1 u 2 v 2 }, z = u + iv D. It is clear that the asymptotic boundary of ψ 1 is a circle in Π { } of radius r, see Figure 2. Example 2 For ε = 1/4, let ψ : D H 3 be the BLW-surface associated with the Weierstrass data 2z, 3dz/81 z 2, z D. Since a solution of the equation 2{G, z} = 3/1 z 2 is the holomorphic map Gz = 1 z 2 /4, z D, we conclude from Theorem 4 and 32 that the immersion ψ is regular. Consider the new conformal parameter x+iy = ϕz = 2i/1 z and take τ the isometry of H 3 given by the Euclidean inversion with respect to the unit sphere centered at 0, 0, 0. Then τ ψ is parametrized as ψ 2 ϕ, where ψ 2 x, y = { 1 4 y + 8 y3 + 8 x 2 5 x 4 + y 2 5 3 x 2, D 2 x 6 y 2 + 2 y 3 + 2 x 2 + 3 y 1 + x 2, 8 1 + y y 2 + x 2 3 2 D D and D := 1 12 y 5 +8 y 6 +11 x 2 x 4 +5 x 6 +y 3 16 24 x 2 +3 y 4 3 + 7 x 2 +2 y + 3 y x 2 2 4 y 1 + 3 x 4. It is clear, see Figure 3, that the immersion τ ψ is regular but its asymptotic boundary is a non regular Jordan curve. Example 3 Let ε = 1/4n 2 + 4n, with n a positive integer and ψ : D H 3 be the BLWsurface associated with the Weierstrass data z/ ε, n+1 2n εdz/1 z 2 2, z D. Since a solution of the equation n{g, z} = n + 1/1 z 2 2 is the meromorphic map Gz = tan 1 + 2 n π + 2 i log 1+z 1 z, 4 z D, it is not difficult to see that the immersion is not embedded and has as asymptotic boundary the set {0, x 2, 0 : x 2 R} { } in Π { }. In fact, for n = 1 if we consider τ the isometry of H 3 given by the Euclidean inversion with respect to the unit sphere centered at 1, 1, 0, then the asymptotic boundary of τ ψ is the }, 18

circle of radius 1/2 centered at 1/2 + i on Π { }. Moreover, by taking the new conformal parameter x + iy = ϕz = log 1+z 1 z, the immersion τ ψ is parametrized as ψ 3 ϕ, where ψ 3 x, y = 1 {4 cosy 4 cos3 y + cos5 y + 2 13 + 4 cos2 y cosh3 x sinh3 x, B 2 4 cosy 4 cos3 y + cos5 y + 13 + 4 cos2 y 2 cosh3 x sinh3 x, 12 2 cosy}, and B = 8 cosy 8 cos3 y+2 cos5 y+3 13 + 4 cos2 y cosh3 x 4 siny+4 sin3 y sin5 y 2 13 + 4 cos2 y sinh3 x, see Figure 4. Example 4 We fix ε = 1/8 and take the BLW-surface ψ : D H 3 associated with the Weierstrass data 2 2z, 5/2 21 z 2 2 dz, z D. Since a solution of the equation {G, z} = 10/1 z 2 2 is the holomorphic map Gz = tanlog z + 1 z 1, z D, from Theorem 4 and 32 ψ is a regular immersion and its asymptotic boundary are two disjoint circles in Π { }. By taking the new conformal parameter x + iy = ϕz = log z+1 z 1, the immersion ψ is parametrized as ψ 4 ϕ, where 13 + 5 cos2 y sin2 x ψ 4 x, y = { cos2 x 13 + 5 cos2 y 3 7 + cos2 y cosh2 y 4 sin2 y sinh2 y, 4 cosh2 y sin2 y + 3 7 + cos2 y sinh2 y cos2 x 13 5 cos2 y + 3 7 + cos2 y cosh2 y + 4 sin2 y sinh2 y, 16 2 cosy cos2 x 13 + 5 cos2 y 3 7 + cos2 y cosh2 y 4 sin2 y sinh2 y }, It is clear that the immersion can be considered as a embedding of a cylinder with two disjoint circles in Π { } as its asymptotic boundary, see Figure 5. For the study of these surfaces at infinity we pose the following Plateau problem: Π { }, find a complete BLW- Given ε 0 < 0 and a Jordan curve Γ on S 2 surface ψ : S H 3 satisfying with Γ as its asymptotic boundary. 2 ε 0 H 1 + ε 0 1K I = 0 19

Since every preserving orientation isometry of H 3 induces a Möbius transformation on S 2 see, for instance, [24] then we will assume that, up to a Möbius transformation, the Jordan curve Γ lies on Π. Proposition 1 Let Γ be a Jordan curve on Π {x 3 = 0} R 3, intγ the bounded component of Π\Γ and G : D I Γ a conformal equivalence where I Γ = {x 1 +ix 2 C : x 1, x 2, 0 intγ}. Then, if G satisfies 27 for a fixed ε < 0, the immersion ψ : D H 3 associated with G by Theorem 4 lies in the region R 3 and it is a solution to the Plateau problem for the Jordan curve Γ. Moreover, 1. ψ has a continuous extension ψ to the closure D of D, such that ψ : D Γ is a D homeomorphism. If Γ is a C smooth Jordan curve then ψ and its derivatives have continuous extensions to D, 2. if ψ is an embedding and Γ is a differentiable curve such that each tangent line to Γ does not pass through a fixed point p intγ then the bounded domain by ψd intγ, O, is starshaped from p in R 3. Proof : First, we observe that G has an extension to a homeomorphism G of the closed disk D onto the closure of I Γ [8, Theorem II.3.4]. Moreover, if Γ is differentiable the above extension is a diffeomorphism [8, Theorem II.3.5]. On the other hand, from 28 and 32, the immersion can be written as 34 ψz = G + ε G z 1 z 2 A 1 ε A 2, ε G z 1 z 2 1 ε A 2. Thus, if we consider the extension ψ of ψ in R 3 such that ψz = Gz, 0 when z = 1, ψ is a continuous map on D since 1 ε A 2 1, A is bounded 1 ε A 2 and G z 1 z 2 0 when z 1 see, [4, Lemma 14.2.8]. In particular, ψ is a solution to the Plateau problem for the Jordan curve Γ. Moreover, if Γ is differentiable then G is also differentiable and G z 0 for z = 1, so A is well defined on D and ψ is a smooth extension of ψ in R 3. Now, let us assume that ψ is an embedding and that Γ is a differentiable curve such that each tangent line to Γ does not pass through a fixed point p intγ. We consider, B t = {q R 3 : q p t} {x 3 0}, B + t = {q R 3 : q p t} {x 3 0} 20

where. denotes the usual Euclidean norm and H t = Bt B t + the closed Euclidean hemisphere with radius t and centered at p. The Euclidean inversion I t with respect to H t is a hyperbolic reflection when I t is restricted to H 3 and so we can use the Alexandrov reflection principle. That is, let Σ t = ψd Bt, Σ+ t = I t ψd B t +. Since ψd is compact there exists t 0 such that ψd B t 0. Therefore, in a standard way, if O was not starshaped from p then, when the radius decreases from t 0 to 0, there would exist a first time t 1 such that either Σ t 1 and Σ + t 1 are tangent at a point in their boundaries or they meet at a interior point. These situations cannot happen for a point of Γ because ψ is differentiable and all tangent lines to Γ do not pass through p. Then, by using that 3 is an elliptic equation we have that Σ t 1 and Σ + t 1 would be the same surface which is a contradiction. Remark 6 From Remark 5, the solution to the Plateau problem associated with a Jordan curve Γ given by the above proposition does not depend on the selected conformal equivalence G from D onto I Γ. Moreover, if ψ is an embedding, the hyperbolic normal to the immersion points towards O. Otherwise, the hyperbolic normal would point upwards at the Euclidean highest point p 0 of the immersion, but the vertical straight half lines are hyperbolic geodesics and so Gp 0 = which contradicts that GD I Γ. First, we solve the Plateau problem at infinity when the Jordan curve is convex. Theorem 5 Let Γ be a convex Jordan curve on Π {x 3 = 0} R 3. Then for any ε < 0 there exists a unique embedded solution to the Plateau problem for Γ with hyperbolic normal pointing downwards at its Euclidean highest point. Moreover if 1/2 < ε < 0, then there are exactly two embedded solutions to the Plateau problem for Γ. Proof : Let us consider the bounded component intγ of Π \ Γ and a conformal equivalence G from D onto I Γ = {x 1 + ix 2 C : x 1, x 2, 0 intγ}. Then, from [14] or [9, Corollary 3], we obtain {G, z} 2/1 z 2 2 < 21 ε/ ε1 z 2 2, ε < 0. Therefore, from Proposition 1, the immersion ψ associated with ε and G is a solution of the Plateau problem for Γ. Now, we shall prove that ψ is an embedding. We observe that the induced metric I of the immersion ψ is conformal to its usual metric I 0 as an immersion into R 3, in fact, from 33, I 0 = ψ 2 3 I where ψ 3 > 0 denotes the third coordinate immersion. Hence, the Euclidean curvature K 0 of ψ can be calculated as ψ 2 3 K 0 = K I I log ψ 3, 21

here I denotes the Laplacian associated with the induced metric I. Since the Weierstrass data of the immersion ψ are given by hz = z/ ε and ω = 1/2 ε{g, z}dz we obtain from 19, 23, 34 and by a straightforward computation of the above Laplacian that and K I = 16ε ε 2 {G, z} 2 1 z 2 4 41 ε 2 K 0 = 4ε 41 A 2 2 {G, z} 2 1 z 2 4 G z 2 1 z 2 2 41 ε 2 ε 2 1 z 2 4. Thus, by using [9, Corollary 3], one has 41 A 2 2 {G, z} 2 1 z 2 4 0 and ψ is an immersion with non negative Euclidean curvature and planar boundary, that is, ψ is an embedding see, for instance, [7] or [27]. From Remark 6, the hyperbolic normal of the immersion ψ points downwards at its Euclidean highest point. Let us assume χ : S H 3 R 3 is another solution to the Plateau problem for a fixed ε 0 < 0 and the convex Jordan curve Γ, with hyperbolic normal pointing downwards at its highest point, or equivalently, pointing towards the enclosed Euclidean domain by clχs intγ. Without loss of generality we may assume the origin belongs to intγ. Let C 0 and C 2 be two circles on Π centered at the origin and bounding a closed annulus A in Π containing Γ in its interior. Choose C 0 and C 2 so that the totally umbilical BLW-surfaces spherical caps, with hyperbolic normal pointing downwards, S 0 and S 2 associated with ε 0 and asymptotic boundary C 0 and C 2, respectively, satisfy S 0 is below χs and χs is below S 2 see Example 1. Then, we consider Γ t, 0 t 2, a foliation of the annulus A such that Γ t is convex for all t [0, 2], Γ 0 = C 0, Γ 1 = Γ and Γ 2 = C 2. Let G t be the conformal equivalence from D onto {x 1 + ix 2 C : x 1, x 2, 0 intγ t } such that G t 0 = 0 and G t z 0 is a positive real number. Since Γ t is also convex we have {G t, z} 2/1 z 2 2 see [14] and we can consider, from Proposition 1, the BLW-surface ψ t : D H 3 R 3 associated with ε 0 and hyperbolic Gauss map G t. In particular, ψ 0 D = S 0, ψ 1 D = ψd and ψ 2 D = S 2. Moreover, ψ t is an embedding as it was proved for ψ. We want to show that χs = ψ 1 D. First, we observe that if t n [0, 2] and {t n } is a sequence converging to t 0 then { G tn } converges uniformly to G t0 see [8, Theorem II.5.2], where G t is the extension of G t to D. Let us prove that if 0 t < 1 then ψ t D is below χs. If J = {t [0, 1[: ψ t D χs } is non empty then we see that J has a minimum; indeed we observe that given a sequence {t n } J converging to the infimum t 0 of J, there would exist p n D such that ψ tn p n clχs and a subsequence {p m } would converge to a point p 0 D and, since {ψ tm } converges uniformly to ψ t0 then ψ t0 p 0 clχs. But, p 0 / D because of Γ t0 Γ 1 =, that is, t 0 J. 22

Assume J then, using the uniform convergence for any t [0, t 0 [, one has that ψ t and ψ t0 are below χs. Thus, since ψ t0 D and χs are tangent at some point with the same hyperbolic normal and ψ t0 D is below χs we have that ψ t0 D and χs agree, but this is not possible because they have different asymptotic boundary. Consequently, J = and using another time the uniform convergence one has ψ 1 D is below χs. Analogously, it can be showed that χs is below ψ 1 D reasoning for the interval [1, 2] and it yields ψ 1 D = χs. Let us now consider ε 0 ] 1/2, 0[, we want to prove that there exists another embedded solution to the Plateau problem for ε 0 and for the convex Jordan curve Γ. Let Γ t, 0 t 1, be a foliation of the topological annulus bounded by the circle C 0 and Γ in Π such that Γ t is smooth for any t [0, 1[, Γ 0 = C 0 and Γ 1 = Γ. Let f : R 3 \ {0, 0, 0} R 3 \ {0, 0, 0} be the map given by x 1 x 2 x 3 35 fx 1, x 2, x 3 = x 2 1 + x2 2 +, x2 3 x 2 1 + x2 2 +, x2 3 x 2 1 + x2 2 +, x2 3 f is a preserving orientation hyperbolic isometry when it is restricted to H 3. Bearing in mind that the origin belongs to intγ t for any t [0, 1], we take the Jordan curves ϕ t = fγ t Π non necessarily convex. And we also call G t to the conformal equivalence from D onto {x 1 + ix 2 : x 1, x 2, 0 intϕ t } such that G t 0 = 0 and G t z 0 is a positive real number. Then, from [13], one has {G t, z} 6/1 z 2 2 and, using that 1/2 < ε 0 and Proposition 1, the immersion ψ t : D H 3 associated with ε 0 is well defined and ψ t is a solution to the Plateau problem for the Jordan curve ϕ t, or equivalently, f ψ t is a solution to the Plateau problem for Γ t since f f is the identity. Moreover, if ψ t is an embedding with hyperbolic normal pointing downwards at its highest point then f ψ t is an embedding with hyperbolic normal pointing upwards at its highest point because the origin is in intϕ t. We want to prove that ψ 1 is an embedding. If we define the interval L = {t [0, 1] : ψ s is an embedding for all s [0, t]}, we know that 0 L and only need to prove that L is open and closed in [0, 1]. We claim L must be open. Otherwise, since L is an interval, L = [0, t 0 ] with t 0 1 and there would exist a sequence {t n } ]t 0, 1[ converging to t 0 such that ψ tn is not an embedding. Hence, there would exist p n, q n D, p n q n, satisfying ψ tn p n = ψ tn q n. But a subsequence {p m } of {p n } must converge to a point p 0 D and a subsequence {q r } of {q m } converges to a point q 0 D. Reasoning as above, { ψ tr } converges uniformly to ψ t0, where ψ t denotes the differentiable extension of ψ t to D. In that way, ψ t0 p 0 = ψ t0 q 0, but ψ t0 is an embedding, that is, p 0 = q 0. From the uniform convergence, there exists a neighbourhood U of p 0 = q 0 in D and r 0 such that ψ tr U is a graph on the tangent plane to the immersion ψ t0 at p 0 = q 0 for all r r 0, which contradicts that ψ tr p r = ψ tr q r with p r q r for r big enough such that p r, q r U. 23

L must be closed. We observe that the origin is a point in intϕ t for any t [0, 1] and the tangent line l to ϕ t at a point p does not pass through the origin, otherwise, we would obtain that the tangent line to Γ t = fϕ t at fp is fl = l which is a contradiction because Γ t is convex and the origin is in intγ t. Consequently, if the interval L is not closed then L = [0, t 0 [ and one has, from Proposition 1, that the bounded Euclidean domain O t enclosed by ψ t D intγ t is starshaped from the origin for all t L. Using again the uniform convergence of ψ t for t t 0, the domain O t0 enclosed by ψ t0 D intγ t0 is also starshaped from the origin and then ψ t0 must be an embedding. Hence, L = [0, 1] and ψ 1 is an embedding. The uniqueness of f ψ 1 as the only embedding with asymptotic boundary Γ and hyperbolic normal pointing upwards at its highest point can be proven in a similar way to the above case. Remark 7 Notice that the Euclidean homotheties with center at a point of Π are preserving orientation hyperbolic isometries when they are restricted to H 3 R 3 see [24] and then the uniqueness part of the above theorem could be proven in an easier way changing the above foliation and considering the foliation given by the image of ψd using the homotheties with center at a point in intγ. Though this easier proof can be applied when intγ is starshaped, it does not work for a general Jordan curve as it will be needed in Theorem 6. Finally, we shall consider the case of a general Jordan curve Γ. Theorem 6 Let Γ be a Jordan curve on Π {x 3 = 0} R 3. Then for any ε ] 1/2, 0[ there exist at least two solutions to the Plateau problem for Γ. Moreover, if 1/4 < ε < 0 then there only exist two embedded solutions to this Plateau problem. Proof : Without loss of generality we can assume the origin belongs to intγ, thus reasoning as in Theorem 5, if we take ϕ = fγ, where f is given by 35, and G 1 : D {x 1 + ix 2 : x 1, x 2, 0 intγ} G 2 : D {x 1 + ix 2 : x 1, x 2, 0 intϕ} are two conformal equivalences then, from [13], {G i, z} 6/1 z 2 2, for i = 1, 2 and, using ε ] 1/2, 0[ and Proposition 1, we can consider the immersions ψ 1, ψ 2 : D H 3 R 3 with associated hyperbolic Gauss map G 1 and G 2, respectively. Therefore, ψ 1 and f ψ 2 are two solutions to the Plateau problem for Γ. On the other hand, if 1/4 < ε < 0 then ψ 1 and ψ 2 must be embedded. In fact, we shall prove that ψ 1 is a graph on the plane Π and analogously could be done for ψ 2. From 29 and 32, the hyperbolic normal η of ψ 1 is given by η = 2ε G 1 z 1 z 2 A 1 ε A 2 2, ε G 1 z 1 z 2 1 + ε A 2 1 ε A 2 2. 24

And, since from 33, the induced metrics on ψ 1 from R 3 and H 3 are conformal then the standard Euclidean normal N to ψ 1 is N = 2 ε G 1 z A + ε A 2, 1 G 1 z 1 ε A 2 1 ε A 2. On the other hand, A 2 because of {G 1, z} 6/1 z 2 2 see [16, Folgerung 2.3] and the third coordinate of N is negative. Consequently, the projection p : ψ 1 D intγ given by px 1, x 2, x 3 = x 1, x 2 is a local diffeomorphism and it is proper, therefore, p is a global diffeomorphism and ψ 1 D is a graph on the plane Π. By considering a foliation Γ t, 0 t 2, of an annulus A centered at a point in intγ such that Γ 0 is a small enough circle in intγ, Γ 1 = Γ and Γ 2 is a big enough circle containing Γ in its interior, one can prove the uniqueness part in a similar way to Theorem 5, bearing in mind that any immersion ψ t associated with Γ t is an embedding. Remark 8 A solution to the Plateau problem for a non regular Jordan curve can be found in Example 2. Moreover, from Example 3, we can see that the condition of embeddedness in Theorems 5 and 6 is essential. References [1] J. A. Aledo and J. A. Gálvez, Complete Surfaces in the Hyperbolic Space with a Constant Principal Curvature, to appear in Math. Nachr. [2] R. L. Bryant, Surfaces of mean curvature one in hyperbolic space, Astérisque 154-155 1987, 321 347. [3] P. Collin, L. Hauswirth and H. Rosenberg, The geometry of finite topology Bryant surfaces, Ann. of Math. 153 2001, 623 659. [4] J. B. Conway, Functions of one complex variable II, Springer-Verlag, New York, 1995. [5] M. P. do Carmo and H. B. Lawson, On Alexandrov-Bernstein theorems in hyperbolic space, Duke Math. J. 50 1983, 995 1003. [6] J. A. Gálvez, A. Martínez and F. Milán, Flat surfaces in the hyperbolic 3-space, Math. Ann. 316 2000, 419 435. [7] M. Ghomi, Gauss map, topology, and convexity of hypersurfaces with nonvanishing curvature, Topology 41 2002, 107 117. [8] G. M. Goluzin, Geometric theory of functions of a complex variable, American Mathematical Society, Providence, R.I. 1969. [9] R. Harmelin, Locally convex functions and the Schwarzian derivative, Israel J. Math. 67 1989, 367 379. 25

[10] M. Kokubu, M. Umehara and K. Yamada, An elementary proof of Small s formula for null curves in PSL2,C and an analogue for Legendrian curves in PSL2,C, to appear in Osaka J. Math. [11] N. J. Korevaar, R. Kusner, W. H. Meeks and B. Solomon, Constant mean curvature surfaces in hyperbolic space, Amer. J. Math. 114 1992, 1 43. [12] G. Levitt and H. Rosenberg, Symmetry of constant mean curvature hypersurfaces in hyperbolic space, Duke Math. J. 52 1985, 53 59. [13] Z. Nehari, The Schwarzian derivative and schlicht functions, Bull. Am. Math. Soc. 55 1949, 545 551. [14] Z. Nehari, A property of convex conformal maps, J. Analyse Math. 30 1976, 390 393. [15] F. Pacard and F. Pimentel, Attaching handles to Bryant surfaces, preprint available at http://arxiv.org/math.dg/0112224. [16] C. Pommerenke, Linear-invariante Familien analytischer Funktionen I, Math. Ann. 155 1964, 108 154. [17] H. Rosenberg and R. Sa Earp, The Geometry of Properly Embedded Special Surfaces in R 3 ; e.g., Surfaces Satisfying ah + bk = 1, where a and b are Positive, Duke Math. J. 73 1994, 291 306. [18] H. Rosenberg and J. Spruck, On the existence of convex hypersurfaces of constant Gauss curvature in hyperbolic space, J. Differential Geom. 40 1994, 379 409. [19] R. Sa Earp and E. Toubiana, Sur les surfaces de Weingarten spéciales de type minimal, Bol. Soc. Brasil. Mat. 26 1995, 129 148. [20] R. Sa Earp and E. Toubiana, Classification des surfaces de type Delaunay, Amer. J. Math. 121 1999, 671 700. [21] R. Sa Earp and E. Toubiana, Symmetry of Properly Embedded Special Weingarten Surfaces in H 3, Trans. Am. Math. Soc. 351 1999, 4693 4711. [22] W. Rossman, M. Umehara and K. Yamada, Mean curvature 1 surfaces in hyperbolic 3-space with low total curvature I, preprint available at http://arxiv.org/math.dg/0008015. [23] A. J. Small, Surfaces of constant mean curvature 1 in H 3 and algebraic curves on a quadric, P. Am. Math. Soc. 122 1994, 1211 1220. [24] M. Spivak, A comprehensive introduction to Differential Geometry, Vol 4, Publish or Perish, Inc., Berkeley, 1979. [25] M. Umehara and K. Yamada, Complete surfaces of constant mean curvature 1 in the hyperbolic 3-space, Ann. of Math. 137 1993, 611 638. [26] M. Umehara and K. Yamada, A parametrization of the Weierstrass formulae and perturbation of complete minimal surfaces in R 3 into the hyperbolic 3-space, J. Reine Angew. Math. 432 1992, 93 116. [27] J. van Heijenoort, On locally convex manifolds, Comm. Pure Appl. Math. 5 1952, 223 242. 26

Departamento de Geometría y Topología Facultad de Ciencias Universidad de Granada 18071 Granada. Spain. email: jagalvez@goliat.ugr.es, amartine@goliat.ugr.es, milan@goliat.ugr.es Figure 1. 27

Figure 2: Totally umbilical BLW-surfaces. Figure 3: BLW-surface with non regular asymptotic boundary. 28

Figure 4: Non embedded BLW-surface with embedded asymptotic boundary. Figure 5: BLW-surface with a non connected embedded asymptotic boundary. 29